tl;dr: Contra my previous opinion, controllers are Service objects; my understanding of a "Service", regarding DI/SL Containers, was incorrect. However, I do continue to opine that it is better use a Factory to get new Controller instances, rather than getting them from the Container directly in a non-Factory object. All that and more in this followup post.
I thought of Services in a Container as objects that are retained for reuse throughout the application; e.g. a PDO instance, a logger, etc.
ahundiak disagreed, and pointed to the Symfony explanation of a Service as "objects that do something", even when not shared throughout the system.
I found that explanation unsatisfying and not well-defined.
However, ahundiak also said "Symfony only creates service definitions for objects which are actually injected somewhere." That piqued my interest, because it got closer to the heart what I was trying to express.
From there we ventured a bit further afield.